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Applied Mathematics at Oxford Christian Yates Centre for Mathematical Biology Mathematical Institute

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Who am I? Completed my B.A. (Mathematics) and M.Sc. (Mathematical Modelling and Scientific Computing) at the Mathematical Institute as a member of Somerville College. Currently completing my D.Phil. (Mathematical Biology) in the Centre for Mathematical Biology as a member of Worcester and St. Catherines colleges. Next year – Junior Research Fellow at Christ Church college. Research in cell migration, bacterial motion and locust motion. Supervising Masters students. Lecturer at Somerville College Teaching 1st and 2nd year tutorials in college.

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Outline of this talk The principles of applied mathematics A practical example Mods applied mathematics (first year) Celestial mechanics Waves on strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves on strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Calculus of variations Mathematical Biology Reasons to study mathematics

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Principles of applied mathematics Start from a physical or real world system Use physical principles to describe it using mathematics For example, Newtons Laws Derive the appropriate mathematical terminology For example, calculus Use empirical laws to turn it into a solvable mathematical problem For example, Law of Mass Action, Hookes Law Solve the mathematical model Develop mathematical techniques to do this For example, solutions of differential equations Use the mathematical results to make predictions about the real world system

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Simple harmonic motion Newtons second law Force = mass x acceleration Hookes Law Tension = spring const. x extension Resulting differential equation

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simple harmonic motion Re-write in terms of the displacement from equilibrium which is the description of simple harmonic motion The solution is with constants determined by the initial displacement and velocity The period of oscillations is

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Putting maths to the test: Prediction At equilibrium (using Hookes law T=ke): Therefore: So the period should be:

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Experiment Equipment: Stopwatch Mass Spring Clampstand 1 willing volunteer Not bad but not perfect Why not? Air resistance Errors in measurement etc Old Spring Hookes law isnt perfect etc

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves on strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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Celestial mechanics Newtons 2nd Law Newtons Law of Gravitation The position vector satisfies the differential equation Solution of this equation confirms Keplers Laws

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How long is a year? M=2x10 30 Kg G=6.67x10 -10 m 3 kg -1 s -2 R=1.5x10 11 m Not bad for a 400 year old piece of maths. Kepler

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves on strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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Waves on a string Apply Newtons Laws to each small interval of string... The vertical displacement satisfies the partial differential equation Known as the wave equation Wave speed:

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Understanding music Why dont all waves sound like this? Because we can superpose waves on each other =

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By adding waves of different amplitudes and frequencies we can come up with any shape we want: The maths behind how to find the correct signs and amplitudes is called Fourier series analysis. Fourier series

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More complicated wave forms Saw-tooth wave: Square wave:

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves of strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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Fluid mechanics Theory of flight - what causes the lift on an aerofoil? What happens as you cross the sound barrier?

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves of strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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Classical mechanics Can we predict the motion of a double pendulum? In principle yes. In practice, chaos takes over.

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves of strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Mathematical Biology Reasons to study mathematics

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How we do mathematical biology? Find out as much as we can about the biology Think about which bits of our knowledge are important Try to describe things mathematically Use our mathematical knowledge to predict what we think will happen in the biological system Put our understanding to good use

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Mathematical biology

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Locusts

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Switching behaviour Locusts switch direction periodically The length of time between switches depends on the density of the group 30 Locusts 60 Locusts

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Explanation - Cannibalism

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Outline of this talk The principles of applied mathematics A simple example Mods applied mathematics (first year) Celestial mechanics Waves on strings Applied mathematics options (second and third year) Fluid mechanics Classical mechanics Calculus of variations Mathematical Biology Reasons to study mathematics

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Why mathematics? Flexibility - opens many doors Importance - underpins science Ability to address fundamental questions about the universe Relevance to the real world combined with the beauty of abstract theory Excitement - finding out how things work Huge variety of possible careers Opportunity to pass on knowledge to others Me on Bang goes the theory

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Im off to watch Man City in the FA cup final

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Further information Studying mathematics and joint schools at Oxford http://www.maths.ox.ac.uk David Achesons page on dynamics http://home.jesus.ox.ac.uk/~dacheson/ mechanics.html http://home.jesus.ox.ac.uk/~dacheson/ mechanics.html Centre for Mathematical Biology http://www.maths.ox.ac.uk/groups/math ematical-biology/ http://www.maths.ox.ac.uk/groups/math ematical-biology/ My web page http://people.maths.ox.ac.uk/yatesc/

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Simple Harmonic Motion This type of motion is the most pervasive motion in the universe. All atoms oscillate under harmonic motion. We can model this motion.

Simple Harmonic Motion This type of motion is the most pervasive motion in the universe. All atoms oscillate under harmonic motion. We can model this motion.

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